Recursive Universes and Emergent Life: A Mathematical Framework for Infinite Hierarchies

Could a universe be one member of an unbounded hierarchy in which each level generates or simulates a universe below it? This paper develops a mathematical and conceptual framework for investigating this possibility and its connections to emergence, life-like organization, intelligence, interlevel communication, and termination risk. Each universe is represented by a state space, transition rule, parameters, initial conditions, and observables. Different levels may have different physical laws, constants, dimensions, and interpretations, while abstract mathematical and logical structures provide a possible common descriptive language. The paper distinguishes representational models from claims about physical realization, formalizes observational equivalence, and identifies sufficient closure conditions for defining an infinite sequence of universe models. It adopts successful generation at every level (\(p_i = 1\)) as an idealized working assumption and examines possible motivations for recursive generation and preservation, including demonstrating the feasibility of universe simulation, scientific discovery, and the value assigned to descendants. It also considers how termination might influence decisions across levels or, under specific implementation assumptions, threaten the stability of a host system. Resource constraints are treated as implementation-dependent rather than universal properties of simulated universes. The framework does not establish that an infinite physical hierarchy exists, that our universe is simulated, or that life necessarily emerges from transformation rules. Its contribution is to clarify the assumptions underlying infinite-chain arguments, distinguish mathematical recursion from physical realization, and identify the theoretical and empirical questions that must be resolved before such a hierarchy can be regarded as a physically supported possibility.